# Matrix Geometric Solutions

## Abstract

In Example 8.12 we analyzed a scalar state process that was a modification of the M/M/1 queue. In the example, we classified two sets of states: boundary states and repeating states. Transitions between the repeating states had the property that rates from states 2, 3,..., were congruent to rates between states *j*, *j* + 1,... for all *j* ≥ 2. We noted in the example that this implied that the stationary distribution for the repeating portion of the process satisfied a geometric form. In this chapter we generalize this result to vector state processes that also have a repetitive structure. The technique we develop in this chapter to solve for the stationary state probabilities for such *vector state* Markov processes is called the *matrix geometric method.* (The theory of matrix geometric solutions was pioneered by Marcel Neuts; see [86] for a full development of the theory.) In much the same way that the repetition of the state transitions for this variation of the M/M/1 queue considered in Example 8.12 implied a geometric solution (with modifications made to account for boundary states), the repetition of the state transitions for vector processes implies a geometric form where scalars are replaced by matrices. We term such Markov processes *matrix geometric processes*.

## Keywords

Markov Process Phase Distribution Rate Matrix State Transition Diagram Main Processor## Preview

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## Bibliographic Notes

- [86]M.F. Neuts.
*Matrix Geometric Solutions in Stochastic Models*. John Hopkins University Press, 1981.Google Scholar - [88]M.F. Neuts.
*Structure of Stochastic Matrices of M/G/1 Type and Their Applications*. Marcel-Deckker, 1990.Google Scholar - [79]L. Lipsky.
*Queueing Theory - A Linear Algebra Approach*. Macmillan, 1992.Google Scholar - [114]H.C. Tijms.
*Stochastic Modelling and Analysis: A Computational Approach*. John Wiley and Sons, 1986.Google Scholar - [74]G. Latouche and V. Ramaswami. A logarithmic reduction algorithm for quasi-birth-and-death processes.
*Journal.of Applied Probability*, 30: 650–674, 1993.MathSciNetMATHCrossRefGoogle Scholar